Unified Time Scale and Continuous Complexity Geometry\\ of Computational Universes:\\ Scattering Mother Scale, Control Manifold, and Construction of Metric G

In previous works, we axiomatized the ``computational universe'' as discrete object U_{comp} = (X,T,C,I), and separately constructed discrete complexity geometry and discrete information geometry on it. However, in that framework, the single-step cost function C remained abstractly assigned, with its connection to real physical time scales not yet systematically characterized. This paper, based on the unified time scale scattering mother scale $ \kappa(\omega) = \varphi'(\omega)/\pi = \rho_{rel}(\omega) = (2\pi)^{-1}\tr\,Q(\omega), introduces ``control manifold'' M and scattering family S(\omega;\theta), systematically embedding the cost of discrete steps in computational universe into a Riemannian-type metric G induced by \kappa(\omega), thereby constructing continuous complexity geometry consistent with physical time scales. Specifically, we first view each physically realizable computational universe U_{comp} as combination of some controllable scattering system: configuration updates are driven by control parameter \theta \in M, scattering matrix S(\omega;\theta) describes physical response in frequency domain, Wigner--Smith group delay matrix Q(\omega;\theta) gives local response of unified time scale density. Subsequently, we define metric G_{ab}(\theta) = \int_{\Omega} w(\omega)\,\tr\big( \partial_a Q(\omega;\theta)\,\partial_b Q(\omega;\theta) \big)\,d\omega and prove: under natural regularity assumptions, G is positive definite with good covariance under control coordinate transformations and internal gauge transformations; furthermore, for any sufficiently smooth control path \theta(t), its length induced by G L_G[\theta] = \int_0^T G_{ab(\theta(t))\,\theta^a(t)\theta^b(t)}\,dt in appropriate discrete limit is equivalent to continuous version of discrete complexity distance. We also prove: for family of computational universes \{U_{comp}^{(h)}\} refined at discrete scale h \to 0, if their single-step costs are constructed from unified time scale scattering response, then configuration graph distance d^{(h)} converges in Gromov--Hausdorff sense to geodesic distance d_G on control manifold. This gives rigorous bridge from completely discrete computational universe to continuous complexity geometry. Finally, we discuss naturality of this continuous complexity geometry in categorical sense: taking control manifold and its metric G as geometric image of ``computational universe objects,'' we can construct category CtrlScat with control--scattering pairs (M,S) as objects, proving existence of functor structure between discrete computational universe category CompUniv and CtrlScat preserving complexity distance. This establishes continuous geometric foundation for subsequently establishing categorical equivalence between ``physical universe category \leftrightarrow$ computational universe category.''